## The modular curve $X_{227}$

Curve name $X_{227}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 3 & 3 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $24$ $X_{84}$
Meaning/Special name
Chosen covering $X_{84}$
Curves that $X_{227}$ minimally covers $X_{84}$, $X_{118}$, $X_{120}$
Curves that minimally cover $X_{227}$ $X_{477}$, $X_{479}$, $X_{488}$, $X_{494}$, $X_{227a}$, $X_{227b}$, $X_{227c}$, $X_{227d}$, $X_{227e}$, $X_{227f}$, $X_{227g}$, $X_{227h}$, $X_{227i}$, $X_{227j}$, $X_{227k}$, $X_{227l}$
Curves that minimally cover $X_{227}$ and have infinitely many rational points. $X_{227a}$, $X_{227b}$, $X_{227c}$, $X_{227d}$, $X_{227e}$, $X_{227f}$, $X_{227g}$, $X_{227h}$, $X_{227i}$, $X_{227j}$, $X_{227k}$, $X_{227l}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{227}) = \mathbb{Q}(f_{227}), f_{84} = \frac{f_{227}}{f_{227}^{2} - \frac{1}{4}}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 + 58950x + 26038750$, with conductor $1530$
Generic density of odd order reductions $25/224$